Suppose that f: ℝ → ℝ is a given measurable function, periodic by 1. For an α ∈ ℝ put Mnαf(x) = 1/n+1 Σk=0nf(x + kα). Let Γf denote the set of those α’s in (0;1) for which Mnαf(x) converges for almost every x ∈ ℝ. We call Γf the rotation set of f. We proved earlier that from |Γf| > 0 it follows that f is integrable on [0; 1], and hence, by Birkhoff’s Ergodic Theorem all α ∈ [0; 1] belongs to Γf. However, Γf\ℚ can be dense (even c-dense) for non-L1 functions as well. In this paper we show that there are non-L1 functions for which Γf is of Hausdorff dimension one.
In this paper, we introduce a new concept of q-bounded radius rotation and define the class R*m(q), m ≥ 2, q ∈ (0, 1). The class R*2(q) coincides with S*q which consists of q-starlike functions defined in the open unit disc. Distortion theorems, coefficient result and radius problem are studied. Relevant connections to various known results are pointed out.
For a Banach space B of functions which satisfies for some m>0
a significant improvement for lower estimates of the moduli of smoothness ωr(f,t)B is achieved. As a result of these estimates, sharp Jackson inequalities which are superior to the classical Jackson type inequality are derived. Our investigation covers Banach spaces of functions on ℝd or for which translations are isometries or on Sd−1 for which rotations are isometries. Results for C0 semigroups of contractions are derived. As applications of the technique used in this paper, many new theorems are deduced. An Lp space with 1<p<∞ satisfies () where s=max (p,2), and many Orlicz spaces are shown to satisfy () with appropriate s.